Minimal cyclic random motion in R n and hyper-Bessel functions
A. Lachal
a,∗, S. Leorato
b, E. Orsingher
baINSA de Lyon, Centre de Mathématiques, Bât. Léonard de Vinci, 20, avenue A. Einstein, 69621 Villeurbanne cedex, France bUniversità di Roma “La Sapienza”, Piazzale A. Moro, 5, 00185 Roma, Italy
Received 6 April 2005; received in revised form 6 October 2005; accepted 8 November 2005 Available online 7 July 2006
Abstract
We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocitycinRn. Then+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with timet), the geometrical features of which are here investigated.
The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of ordern+1. These distributions are proved to be connected with(n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.
Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion inR3with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.
©2006 Elsevier Masson SAS. All rights reserved.
Résumé
Dans ce travail, on étudie l’évolution dans l’espaceRnd’une particule animée d’un mouvement aléatoire cyclique à vitesse constante. Le mouvement est supposé minimal au sens où les différentes directions prises sont au nombre den+1 ; de plus, ces directions forment un hyper-polyèdre régulier fixe. Le support de la distribution de la position de la particule est également un hyper-polyèdre régulier (de taille évolutive au cours du temps).
Faisant appel aux statistiques d’ordre, on a pu obtenir explicitement la loi de probabilité de la position de la particule à un instant donné. Le résultat s’exprime au moyen de fonctions de Bessel généralisées d’ordren+1 et montre que cette étude est liée à des équations aux dérivées partielles hyperboliques d’ordren+1.
Ce travail a été inspiré par une conjecture formulée par Orsingher et Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion inR3with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133], laquelle se révèle finalement être fausse.
©2006 Elsevier Masson SAS. All rights reserved.
MSC:primary 60K99; secondary 62G30, 33E99
Keywords:Cyclic random motions; Hyper-Bessel functions; (n+1)th order partial differential equations; Order statistics; Hyperpolyhedron Mots-clés :Mouvements aléatoires cycliques ; Fonctions de Bessel d’ordren+1 ; Équations aux dérivées partielles d’ordren+1 ; Statistiques
d’ordre ; Hyper-polyèdre
* Corresponding author.
E-mail address:[email protected] (A. Lachal).
0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpb.2005.11.002
1. Introduction
We here study the cyclic motion inRnof a particle (with position(X(t ), t0)) which can taken+1 directions
vj,j=0, . . . , n, and moves with speedc <+∞. The extremities of the vectors representing the directions form a regularn-dimensional polyhedron withn+1 vertices (for short, we say(n+1)-hedron) inscribed in the unit sphere.
The particle can initially choose one of the possible directions with probability n+11. The directions are taken successively at Poisson paced times, that is the particle moving with directionvj, after a Poisson event, takes the directionvj+1,j=0, . . . , n(withvn+1= v0).
The minimality of the number of directions, together with the cyclicity make the derivation of the explicit distrib- utions of motion possible.
Motions of particles in a turbulent medium, for example in the presence of a vortex, can be adequately described by the models studied below.
Usually, the analysis of random motions with finite velocity is either performed by means of analytic arguments, essentially based on partial differential equations, or by a more probabilistic approach, based on order statistics.
The analytic method has been implemented in the study of cyclic motions on the plane (with three directions) and in spaceR3 (with four directions associated with a regular tetrahedron) by Orsingher [6] and Orsingher and Som- mella [7]. Random motions in the plane with three directions and Erlang distributed interarrival times is considered in Di Crescenzo [1].
The approach based on order statistics has proved to be more suitable for generalizations on higher order spaces (as will be applied here) as well as on non-cyclic motions; the cases of planar motions – symmetrically deviating and with uniform choice of directions – is examined in Leorato and Orsingher [4].
This work proves that the conjecture formulated in the paper by Orsingher and Sommella [7] is false and we are now able to obtain the exact distributions of the position of a randomly moving particle inRn and to show that it matches the necessary requirements, including the connection with the partial differential equations governing the probability laws. These equations, derived by different authors (Kolesnik [3], Samoilenko [8,9] and others) are related to hyper-Bessel functions analyzed by Kiryakova [2] and Turbin and Plotkin [10].
Our main result is the derivation of the distribution ofX(t ):
˜
pr( x, t )dx=Pr
X(t )∈dx, complete cycle+rdirections
= dxe−λt (n+1)rn!Vn
λ c
n+r
Hr,n+1( x, t )Ir,n+1
λ c
n+1
n
j=0
hj( x, t )
(1.1)
wherex=(x0, . . . , xn−1), dx=dx0· · ·dxn−1andhj( x, t )=ct+nn−1
i=0vi,jxi=0 are the equations of the hyper- faces of a(n+1)-hedron Tct with volumeVn(ct )n, and(v0,j, . . . , vn−1,j)are the coordinates of the vectorsvj for j=0, . . . , n.
The functionsHr,n+1are defined as Hr,n+1( x, t )= 1
n+1 n j=0
j+r−1 l=j
hl( x, t )
while the hyper-Bessel functionsIr,n(x)are Ir,n(x)=
∞ q=0
1 (q!)n−r((q+1)!)r
x n
nq
, 0rn.
Our paper is organized as follows. The second section is devoted to the geometrical description of the directions of motion together with that of the support of the distributions.
We then turn our attention to the probabilistic analysis of the cyclic motion as well as to the analysis of the related governing equations (Section 3). In particular, the functionsqj( u )for 0j n, defined as
pj( x, t )=const·qj( u )=Pr
X(t )∈dx, the current direction isvj /dx
whereu=(u0, . . . , un)=(h0( x, t ), . . . , hn( x, t )), are governed by the(n+1)th order partial differential equations
∂n+1qj
∂u0· · ·∂un = λ
(n+1)c n+1
qj. (1.2)
We shall see that the p.d.e. (1.2) is related to the(n+1)th order Bessel equation:
w ∂
∂w n+1
q= λw
c n+1
q.
The fourth part of the paper is concerned with the explicit derivation of the conditional probabilities below by means of order statistics:
Pr
X(t )∈dx|N (t )=(n+1)q+r−1
, r=0, . . . , n, q1, (1.3)
whereN (t )denotes the number of Poisson events up to timet.
Our analysis shows that the conditional distributions (1.3) coincide with the corresponding components of (1.12) of Orsingher and Sommella [7] forr=0, nand arbitrary values ofq=1,2, . . ., while forr=1, . . . , n−1 the con- ditional distributions and, a fortiori, the absolutely continuous component differ from those conjectured. By summing (1.1) we get a complete expression for the absolutely continuous part of the distribution of motion in Rn. Let us point out that the singular component of the distribution ofX(t )is spread on then
k=1
n+1
k
subspaces of dimension d=k−1, composing the boundary ofTct, with 0dn−1.
The concluding section is devoted to checking that d
dt
Tct
I0,n+1
λ c
n+1
n
j=0
hj( x, t )
dx
= d
dt Vol(Tct)+
Tct
∂
∂t
I0,n+1
λ c
n+1
n
j=0
hj( x, t )
dx.
For higher-order derivatives, a formula similar to that above does not hold and this is the reason why the conjecture formulated in Orsingher and Sommella [7] is not true.
In Subsection 5.2, we have shown that the distributions obtained (suitably simplified) satisfy an(n+1)th order p.d.e. related to the hyper-Bessel equation.
2. The directional vectors and the geometry ofTct
2.1. The vectors of directions
Letvj,j =0, . . . , n,be the vectors representing the possible directions of the cyclic motion. Letv0=(1,0, . . . ,0) and letvj=(v0,j, v1,j, . . . , vj,j,0, . . . ,0)andvn=(v0,n, v1,n, . . . , vn−1,n). In order to evaluate the numbersvi,j, 0in−1, 0jn, we consider the following symmetry conditions:
vj· vk=const ifj=k, (2.1)
|vj|2=1 for allj , (2.2)
n j=0
vj= 0. (2.3)
The constant in (2.1) is equal to−n1as can be obtained in the following manner:
0= n
j=0
vj
· vk= |vk|2+ n
j=0 j=k
vj· vk=1+nconst.
Thus, the conditions (2.1) and (2.2) can be rewritten as
vj· vk=
−1n ifj =k,
1 ifj =k. (2.4)
The coordinates of the vectorv1=(−1n, v1,1,0, . . . ,0)can immediately be calculated because of (2.2): v1,1=
±
1−n12. By convention we take, throughout the paper, the positive signs. We also get, from (2.4), thatv0,j= −n1, for 1j n, that is, all the first components of the vectorsvj,j1, are identical.
From (2.4) we obtain that v1,j= −1
n n+1
n−1 for 2jn because
−1
n=v0,1v0,j+v1,1v1,j= 1 n2+
√n2−1
n v1,j forj2.
By applying the same procedure we can check that
v2=
−1 n,−1
n n+1
n−1, n+1
n
n−2
n−1,0, . . . ,0
.
This easily permits us to find thatv2,j= −
n+1 n
√ 1
(n−1)(n−2) forj >2 becausev2· vj=2
i=0vi,2vi,j = −1n. We write down the general form of the coordinatesvi,j of the vectorsvj forjn−1
vi,j=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
−
n+1 n
√ 1
(n−i+1)(n−i) ifi < j, n+1
n n−j
n+1−j ifi=j,
0 ifi > j.
(2.5)
The componentsvi,j fori < j can be easily evaluated by induction by using (2.4) as forv1,j andv2,j. Then, in view of (2.2), we find that
vj,j= 1−
j−1
i=0
vi,j2 = 1−
j−1
i=0
n+1 n
1
(n−i+1)(n−i). (2.6)
The sum inside (2.6) can be evaluated without effort:
j−1
i=0
1
(n−i+1)(n−i)=
j−1
i=0
1
n−i − 1
n−i+1
= j
(n+1)(n−j+1). Finally, the last vectorvn, thanks to (2.3) writes
vi,n= − n+1
n
√ 1
(n−i+1)(n−i), for 0in−1.
The matrixVof the components of the directional vectorsvj, j=0, . . . , n, is given in Table 1. The reader can check that conditions (2.3) are fulfilled.
2.2. The(n+1)-hedronTa
Let us introduce the pointsAj,j=0, . . . , n, defined by−−−→OAj=avj, for a fixeda >0. The pointsAj,j=0, . . . , n, are the vertices of a regular(n+1)-hedronTa with centerO. The length of the edges ofTacan clearly be obtained by observing that, for the edge−−−−→AjAk, with endpointsAjandAk, we have that−−−−→AjAk=a(vk− vj)and thus, by (2.4),
−−−−→AjAk2=a2
|vk|2+ |vj|2−2vj· vk
=2n+1 n a2.
Table 1
The matrixVof directions
v0 v1 v2 · · · vn−1 vn
1 −n1 −1n · · · −1n −1n
0
n+1 n n−1
n −1n
n+1
n−1 · · · −1n
n+1
n−1 −1n
n+1 n−1
0 0 (n+1)(n−2)
n(n−1) · · · − n+1
n(n−1)(n−2) − n+1
n(n−1)(n−2) ..
.
.. .
.. .
.. .
.. .
0 0 0 · · · −
n+1
n(n−i+2)(n−i+1) −
n+1 n(n−i+2)(n−i+1) ..
.
.. .
.. .
.. .
.. .
0 0 0 · · · −
n+1 n
1
2·3 −
n+1 n
1 2·3
0 0 0 · · ·
n+1n 1
2 −
n+1n 1 2
2.2.1. Analytic representation ofTa
We now derive the equations of the hyperfaces ofTa, which play an important role in the distribution ofX(t ).
LetAkbe an arbitrary vertex of the hyperfaceFj orthogonal to the vectorvj forj=k. It is clear that
−−−→OM=−−−→OAk+−−−−→AkM=avk+−−−−→AkM, (2.7) whereMis an arbitrary point ofFj (and thus−−−−→AkM is orthogonal tovj) with coordinates(x0, . . . , xn−1). By taking the scalar product of (2.7) byvj we get
n−1
i=0
vi,jxi= vj·−−−→OM= vj·
avk+−−−−→AkM
= −a n.
The equation of the hyperface Fj for j =0, . . . , n is thusn−1
i=0vi,jxi+ an =0. The (n+1)-hedron Ta is then analytically defined as
Ta=
(x0, . . . , xn−1)∈Rn:
n−1
i=0
vi,jxi+a
n0 forj=0, . . . , n
(2.8) and we shall see thatTct is the set of all possible positions of the moving particle at timet.
Remark 2.1.Observe that the inequalityn−1
i=0vi,jxi+an>0 represents the half-space containing the vertexAj= (av0,j, . . . , avn−1,j). Indeed, inAj we have thatan−1
i=0vi,j2 +an=a(1+1n) >0.
2.2.2. Volume ofTa
For our further analysis, it is useful to evaluate the volume of the(n+1)-hedronTa. This can be split up inton+1 not regular(n+1)-hedrons of equal volume obtained fromTawith each vertex successively being replaced by the originO. In this way we have that
Vol(Ta)=(n+1)Vol TOa
(2.9) whereTOa is a(n+1)-hedron with one vertex inO. By means of a well-known formula and by using Table 1, we get
Vol TOa
=an
n! det(v0, . . . ,vn−1)=an n!
1 −1n · · · −n1 0
n+1 n
n−1
n · · · −1n
n+1 n−1
... ... . .. ...
0 0 · · ·
n+1 2n
=an n!
n+1 n
n−1
· 1
√n=(n+1)(n−1)/2 nn/2n! an. Therefore, from (2.9) we have that
Vol(Ta)=(n+1)(n+1)/2
nn/2n! an. (2.10)
In the rest of the paper we shall use the results of this section applied to the case wherea =ct. We also put Vn=Vol(T1)=(n+n1)n/2(nn+!1)/2.
3. Analytic description of random motion 3.1. The support of the random position
We consider the randomly moving pointX(t )=(X0(t ), X1(t ), . . . , Xn−1(t ))which, at timet=0, was at the origin O=(0, . . . ,0)and initially took one of the directionsvj,j=0, . . . , n, with probability n+11.
The motion is cyclic in the sense that at each Poisson event the particle switches from directionvjtovj+1(we set
vn+1= v0and, in general,v(n+1)k+j= vj for any positive integerk).
LetN (t )be the number of Poisson events up to timetand letT1, . . . , Tn, . . .denote the instants where they occur.
Therefore, the positionX(t )of the moving particle starting with directionv0is X(t )=c
I{N (t )1}
N (t )−1 k=0
(Tk+1−Tk)vk+(t−TN (t ))vN (t )
. (3.1)
The first sum in (3.1) refers to the case where at least one change of direction has occurred while the second one is related to the displacement along the current direction at timet. If N (t )=0, thenX(t )=c(t−TN (t ))v0=ctv0 and the particle is located inA0(witha=ct) at timet. If we putTN (t )+1=t the displacement (3.1) takes the form X(t )=cN (t )
k=0(Tk+1−Tk)vk and, by (2.4), for allj =0, . . . , n, X(t )· vj+ct
n =c
N (t )
k=0 vk=vj
−1
n(Tk+1−Tk)+c
N (t )
k=0 vk=vj
(Tk+1−Tk)+ct n
= −c n
N (t )
k=1
(Tk+1−Tk)+c
1+1 n
N (t )
k=0 vk=vj
(Tk+1−Tk)+ct n
=c
1+1 n
N (t )
k=0 vk=vj
(Tk+1−Tk)0.
This permits us to conclude that, in view of (2.8), the moving particle at timet is always inside or on the boundary ofTct.
In order that the moving pointX(t )be located on the boundary∂Tct we must have that X(t )· vj+ct
n =0 for some 0jn.
This equality is realized ifN (t )is such that the identityvk= vj does not hold for any value ofkand thus the set on which the sum is performed is empty. If the initial direction isv0, then
X(t )· vj+ct
n =0 forN (t ) < jn, (3.2)
which means that the moving particle lies on aN (t )-face (face of dimensionN (t )) of∂Tct at timet.
Remark 3.1.For example, forn=2 andN (t )=1 (which impliesj=2 in (3.2)), we have X(t )· v2+ct
2 =0
which represents the side of∂Tct opposite tov2. IfN (t )=0, (3.2) holds forj=1,2 and therefore, the moving point is located simultaneously on the two linesX(t )· v1+ct2 =0 andX(t )· v2+ct2 =0, that is on the vertex(ct,0).
3.2. About the governing equations
LetD(t )be the current direction of motion at timet. ClearlyD(t )takes valuesvj,j=0, . . . , n. For the densities of the probabilitiespj( x, t )dx=Pr{X(t )∈dx, D(t )= vj}forx∈int(Tct)we have the following theorem.
Theorem 3.2.The densitiespjsatisfy the following differential system
∂pj
∂t ( x, t )= −c∂pj
∂vj( x, t )−λ
pj( x, t )−pj−1( x, t )
for0jn, (3.3)
wherep−1=pnand
∂pj
∂vj
( x, t )=−−−→gradpj· vj.
Proof. Suppose that the particle is located inx at timet+t. If no Poisson event has occurred during[t, t+t] (which happens with probability 1−λt+o(t )), then the particle must have been inx−ctvjat timet. If exactly one Poisson event has occurred (with probabilityλt+o(t )) during[t, t+t], the direction of the particle at time t wasvj−1. Finally, the probability that more than one Poisson event has occurred is o(t ). This short discussion leads to the following equality:
pj( x, t+t )=(1−λt )pj( x−ctvj, t )+λtpj−1( x, t )+o(t ).
We next expand
pj( x−ctvj, t )=pj( x, t )−ct
n−1
i=0
∂pj
∂xi( x, t )vi,j+o(t ) and
pj( x, t+t )=pj( x, t )+∂pj
∂t ( x, t )t+o(t ).
Some obvious simplifications and the limit witht→0 yield (3.3). 2 The system (3.3) can be substantially reduced as shown in the next theorem.
Theorem 3.3.Letpj( x, t )=e−λtqj( u )whereuis the vector with components uj=ct+n
n−1
i=0
vi,jxi, j =0, . . . , n. (3.4)
Then the functionsqj satisfy the differential system (n+1)c∂qj
∂uj =λqj−1, j=0, . . . , n. (3.5)
Proof. The exponential transformation applied to (3.3) readily yields the system
∂qj
∂t = −c∂qj
∂vj +λqj−1, j=0, . . . , n. (3.6)
In view of (3.4) we have that
∂qj
∂t =c n k=0
∂qj
∂uk and ∂qj
∂xi =n n k=0
∂qj
∂ukvi,k. (3.7)
Thus, by (2.4),
∂qj
∂vj =−−−→gradqj· vj=
n−1
i=0
∂qj
∂xivi,j=n n k=0
∂qj
∂uk n−1
i=0
vi,jvi,k
=n n k=0 k=j
−1 n
∂qj
∂uk
+n∂qj
∂uj = − n k=0
∂qj
∂uk +(n+1)∂qj
∂uj
. (3.8)
By plugging (3.7) and (3.8) into (3.6) we obtain (3.5). 2
We remark that Eqs. (3.5) have substantially the same form as those of the systems (3.4) of Orsingher and Sommella [7] and (2.8) of Orsingher [6], except for some constants, because of a different definition of the transformation (3.4).
Corollary 3.4.Each functionqj satisfies the following(n+1)th order partial differential equation
∂n+1qj
∂u0· · ·∂un = λ
(n+1)c n+1
qj, j=0, . . . , n. (3.9)
Proof. By differentiating the first equation of (3.5) with respect tounwe get
∂2q0
∂u0∂un = λ (n+1)c
∂qn
∂un = λ
(n+1)c 2
qn−1. (3.10)
By differentiating (3.10) we obtain
∂3q0
∂u0∂un∂un−1= λ
(n+1)c 3
qn−2
and by iterating this procedure we finally get Eq. (3.9) forj=0. The other cases are quite similar. 2
Proposition 3.5.The solutionsq of p.d.e.(3.9)depending only on the variablew= n+√1u0· · ·unverify the(n+1)th order hyper-Bessel equation
w ∂
∂w n+1
q= λw
c n+1
q. (3.11)
Proof. Let us now consider the transformation vj=uj forj=0, . . . , n−1,
w= n+√1
u0· · ·un. (3.12)
In view of (3.12), fori=0, . . . , n−1,
∂qj
∂ui =
n−1
k=0
∂qj
∂vk
∂vk
∂ui +∂qj
∂w
∂w
∂ui =∂qj
∂vi + w
(n+1)vi
∂qj
∂w, (3.13)
and
∂qj
∂un =
n−1
k=0
∂qj
∂vk
∂vk
∂un+∂qj
∂w
∂w
∂un =v0· · ·vn−1 (n+1)wn
∂qj
∂w. (3.14)
In light of (3.13) we have that
∂nqj
∂u0· · ·∂un−1= n−1
i=0
1 vi
vi
∂
∂vi + w n+1
∂
∂w
qj (3.15)
and, by using the well-known formula
n−1 i=0
(αi+x)= n m=0
xm
0l1<···<ln−mn−1
αl1· · ·αln−m=:
n m=0
xmσn−m(α0, . . . , αn−1),
the differential operator in (3.15) can be written as
∂n
∂u0· · ·∂un−1= 1 v0· · ·vn−1
n m=0
w n+1
∂
∂w m
σn−m
v0 ∂
∂v0, . . . , vn−1 ∂
∂vn−1
.
Formula (3.14), applied to the above expression, enables us to write the differential operator in (3.9) as
∂n+1
∂u0· · ·∂un = 1 (n+1)wn
∂
∂w n
m=0
w n+1
∂
∂w m
σn−m
v0 ∂
∂v0, . . . , vn−1 ∂
∂vn−1
= 1 wn+1
n−1
m=0
w n+1
∂
∂w m+1
σn−m
v0
∂
∂v0, . . . , vn−1
∂
∂vn−1
+ 1
((n+1)w)n+1
w ∂
∂w n+1
.
Therefore, the solutions of (3.9) depending only onwsatisfy the ordinary equation (3.11) 2 Remark 3.6.We check here that the hyper-Bessel function
I0,n(w)=∞
k=0
1 (k!)n
w n
nk
is a solution to the hyper-Bessel equation (w∂w∂ )nq =wnq. Since (w∂w∂ )wnk=(nk)wnk and thus(w∂w∂ )nwnk = (nk)nwnk, we have that
w ∂
∂w n
I0,n(w)= ∞ k=0
1 (k!)n
(nk)nwnk
nnk =
∞ k=1
1 nn(k−1)
wnk ((k−1)!)n
=wn ∞ k=0
1 (k!)n
w n
nk
=wnI0,n(w).
This implies that the functionI0,n+1(λcw)is a solution to Eq. (3.11).
4. Order statistics applied to cyclic motions
In this section, we introduce the number Nj(t )of times the direction vj is taken, up to time t. Of course, the random numbersNj(t )andN (t )are linked by
n j=0
Nj(t )=N (t )+1.
We can immediately write down the following relationship:
Pr
X(t )∈dx
=
k0,...,kn0
Pr
N0(t )=k0, . . . , Nn(t )=kn
Pr
X(t )∈dx|N0(t )=k0, . . . , Nn(t )=kn
. (4.1)
For the cyclic motion the explicit form of Pr{N0(t )=k0, . . . , Nn(t )=kn}is almost straightforward and this makes the derivation of (4.1) in a closed form possible.
For the planar motion with three directions, the probabilities Pr{N0(t )=k0, N1(t )=k1, N2(t )=k2} where N0(t )+N1(t )+N2(t )=N (t )+1, have been explicitly evaluated for the symmetrically deviating motion by means of an extension of Bose–Einstein statistics (Leorato and Orsingher [4]).
For a number of directions greater than or equal to 4, the evaluation of the distribution of(N0(t ), . . . , Nn(t )) becomes extremely difficult except for the uniform case, where(N0(t ), . . . , Nn(t ))is a multinomial random vector.
The aim of this section is the evaluation of the conditional probabilities in (4.1) by means of order statistics.
This method has been successfully applied in the case of planar, cyclic motions with orthogonal directions (Leorato et al. [5]) and also for motions with three directions (Leorato and Orsingher [4]).
The results of Subsections 4.1 and 4.2 hold for any form of the chance mechanism regulating the change of direc- tions. Since the behaviour of(N0(t ), . . . , Nn(t ))in the cyclic case is essentially deterministic, the relevant part of the analysis reduces to the derivation of the conditional probabilities appearing in (4.1).
We observe that the minimality of the number of directions is very important in order to obtain the conditional distributions in (4.1) in a relatively simple way as will become clear below.
4.1. Some preliminary results about the position of the particle
We start by writing the vector (3.1) in a new convenient form:
X(t )=c
I{N (t )1} N (t )−1
k=0
(Tk+1−Tk)vk+(t−TN (t ))vN (t )
=c
I{N (t )1} n j=0
0lN (t )−1:
vj is taken in[Tl,Tl+1)
(Tl+1−Tl)vj+ n j=0
(t−TN (t ))vjI{vjis taken in[TN (t),t]}
=ct n j=0
Lj(t )vj (4.2)
where
Lj(t )=1 t
I{N (t )1}
0lN (t )−1:
vj is taken in[Tl,Tl+1)
(Tl+1−Tl)+(t−TN (t ))I{vj is taken in[TN (t),t]}
is the proportion of time spent travelling with directionvj. The r.v.’sLj(t )can also be written as Lj(t )=1
t
Nj(t ) m=1
Tm(j ), (4.3)
whereTm(j )indicates how long the particle has travelled themth time thatvjhas been taken. We remark that n
j=0 Nj(t )
m=1
Tm(j )=t n j=0
Lj(t )=t.
The r.v.’sTm(j ), 1mNj(t ), 0j n, are independent and exponentially distributed with parameterλ.
Theorem 4.1.Fix some positive integersk0, . . . , kn such thatn
j=0kj=k+1. The joint conditional distribution of (L0(t ), . . . , Ln−1(t ))is given by
Pr
L0(t )∈dl0, . . . , Ln−1(t )∈dln−1|N0(t )=k0, . . . , Nn(t )=kn
=f (l0, . . . , ln−1)dl0· · ·dln−1 (4.4) where
f (l0, . . . , ln−1)=n k!
j=0(kj−1)! n j=0
ljkj−1I{l
0,...,ln−10,n−1
j=0lj1}, (4.5)
andln=1−n−1
j=0lj.
